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The search for a person to lead ABC began around the time of the announcement in late January, with potential candidates meeting by phone as well as in Berkshire's hometown of Omaha, Nebraska, and in New York, where J.P. Morgan is based, according to people with knowledge of the process.

Health policy and insurance experts were among the initial targets, with ex-Aetna executive Gary Loveman and former Medicare chief Andy Slavitt on the list, along with Todd Park, who was previously President Obama's technology chief.

More recently, ABC has started looking for a candidate with an entrepreneurial background in technology and health who is far removed from drug supply companies and health plans, which are viewed as part of the problem, said the people, who asked not to be named because the talks are confidential.

One of the top names to emerge in recent weeks was Owen Tripp, the CEO of Grand Rounds Health, a start-up that sells a second medical opinion service as a benefit to large employers like Walmart and Target, the people said. Prior to Grand Rounds, Tripp co-founded Reputation.com, a developer of online reputation software, and he also has a background in health-care consulting.

But Tripp told CNBC in a statement that he's committed to staying on as CEO of Grand Rounds.

"To the extent we can be part of their solution as their mission sharpens, we'd be glad to pitch in — just as we would assist any organization that wants to improve clinical outcomes," Tripp said.

Berkshire CEO Warren Buffett said at the company's investor meeting earlier this month that the consortium hoped to have a CEO "within a couple of months."

Todd Combs, an investment manager at Berkshire and the lead recruiter for ABC, has the challenge of finding a leader who can work across three companies with a combined 1.2 million employees and simultaneously help develop innovative solutions in a multitrillion-dollar industry. The group hasn't said much about how it plans to leverage its size and scale to bring down costs or how it will use technology to simplify the health-care system for consumers.

As if that weren't demanding enough, ABC is structured as an entity "free from profit-making incentives and constraints," according to the initial announcement, although it will not necessarily be a non-profit. That means Combs needs to find someone who can handle the requisite long hours, extensive travel and public scrutiny of running a high-profile start-up, but probably without the allure of a big stock incentive plan, which is typically an attractive incentive in tech recruiting.

"Senior talent with an entrepreneurial background might not be inclined to a structure like that," said Annie Lamont, a managing partner at Oak Investment Partners who invests in health and financial start-ups. "They could find a professional manager, but true entrepreneurs might well be harder."

Longtime technology venture capitalist John Doerr has been tapped to help put forward names from his network. The other executives involved, according to the January announcement, are JPMorgan's Marvelle Sullivan Berchtold and Beth Galetti, a senior vice president at Amazon.

Transcend the challenges

Whoever ultimately takes the top job has to align with Combs, who's been kicking around the idea for the consortium for more than 5 years, according to people familiar with his thinking.

After getting his own company on board, Combs enlisted JPMorgan CEO Jamie Dimon, before bringing on Amazon as the tech partner, said one of the people. The three executives are also known to be friends.

"What's intriguing about this group is the possibility of innovation, tech and a consumer focus coming together to transcend the challenges with the health-care ecosystem," said Brian Marcotte, CEO of the National Business Group on Health, which represents large employers.

One area where industry experts see the consortium having a potentially dramatic impact on pricing is in cutting out the network of middlemen rather than partnering with them, and giving patients more direct access to their medical needs, whether that's in the form of care, coverage or prescriptions.

The group could also expand. One person familiar with the matter said that other large employers have approached ABC about wanting to join once a CEO is announced.

That could be a while, as ABC has yet to find the right fit. Trevor Price, co-founder of executive search firm Oxeon Partners, which specializes in health, said there are significant potential upsides and downsides that make the position a tough sell for some.

"For anyone who believes they can build a fully integrated platform to care for millions of Americans, including the payer and provider and even gets into managing risk, nothing would stop them from taking this job." Price said.

But it's an extremely high profile gig in a historically difficult industry to crack, and it comes with the added weight of working under Jeff Bezos, Warren Buffett and Dimon all at once.

"If the CEO isn't successful," Price said, "it would be hard to come back from that," Price said.